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Multi-parametric Optimization and Control. Efstratios N. PistikopoulosЧитать онлайн книгу.

Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos


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of active inequality constraints is given by images, where images is the number of equality constraints. As the number of critical regions is uniquely defined by the active set, it is bound by above by all possible combinations of active sets, which is given by images, which completes the proof.

      2.1.2 Global Properties

      The solution properties described in Chapter 2.1.1 hold for any feasible point images and thus the following theorem can be formulated:

       Proof

      The two key statements that need to be proven are the convexity of images and images. Consider two generic parameter values images and let images, images and images and images be the corresponding optimal objective function values and optimizers. Additionally, let images and define images and images. Then, since images, images, and images are feasible and satisfy the constraints images and images. As these constraints are affine, they can be linearly combined to obtain images, and therefore images is feasible for the optimization problem (2.2). Since a feasible solution images exists at images, an optimal solution exists at images and thus images is convex.

      The optimal solution at images will be less than or equal to the feasible solution, i.e. images and thus:

      (2.9a)equation

      (2.9b)equation

      i.e. images, images, images. This proves the convexity of images and images. The piecewise affine nature of images and images is a direct result from the fact that the boundary between two regions belongs to both regions. Since the optimum is unique, the optimizer and thus the optimal objective function value must be continuous across the boundary.

      Let each active set images of an mp‐LP problem be a node in the set of solutions images. Then the nodes images and images are connected if (i) there exists a images such that images and images are both optimal active sets and (ii) it is possible to pass from images to images by one step of the dual simplex algorithm. The resulting graph images is fully defined by the nodes images as well as all connections images, i.e. images.


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